Which model can I use for this problem with multiple inputs and outputs?
This depend on type of data you use. 

Time sequence data

If the data advanced in time, a LSTM or similar RNN should be used. RNN calculate output through time. It works very good on time series data as it have a real sense of time. While CNN and MLP could work for time series data, it often don't work that well as different timestep of data is not defined. 

Non- Time sequence data

According to you previous comment, the data seems to be of this kind. In the case of your data, a normal Multi-layer perceptron works well for this. The data is a direct mapping between the input and the output. If the input data is an image, use a CNN. 

For example code in keras, see here:
You need the pandas module for this to work. Run pip install pandas to install pandas.
    from keras.models import Sequential
    from keras.utils import np_utils
    from keras.layers.core import Dense, Activation, Dropout

import pandas as pd
import numpy as np

# Read data
train = pd.read_csv('../input/train.csv')
labels = train.ix[:,0].values.astype('int32')
X_train = (train.ix[:,1:].values).astype('float32')
X_test = (pd.read_csv('../input/test.csv').values).astype('float32')

# convert list of labels to binary class matrix
y_train = np_utils.to_categorical(labels) 

# pre-processing: divide by max and substract mean
scale = np.max(X_train)
X_train /= scale
X_test /= scale

mean = np.std(X_train)
X_train -= mean
X_test -= mean

input_dim = X_train.shape[1]
nb_classes = y_train.shape[1]

# Here's a Deep Dumb MLP (DDMLP)
model = Sequential()
model.add(Dense(128, input_dim=input_dim))
model.add(Activation('relu'))
model.add(Dropout(0.15))
model.add(Dense(128))
model.add(Activation('relu'))
model.add(Dropout(0.15))
model.add(Dense(nb_classes))
model.add(Activation('softmax'))

# we'll use categorical xent for the loss, and RMSprop as the optimizer
model.compile(loss='categorical_crossentropy', optimizer='rmsprop')

print("Training...")
model.fit(X_train, y_train, nb_epoch=10, batch_size=16, validation_split=0.1, show_accuracy=True, verbose=2)

print("Generating test predictions...")
preds = model.predict_classes(X_test, verbose=0)

def write_preds(preds, fname):
    pd.DataFrame({"ImageId": list(range(1,len(preds)+1)), "Label": preds}).to_csv(fname, index=False, header=True)

write_preds(preds, "keras-mlp.csv")


Code source: https://www.kaggle.com/fchollet/simple-deep-mlp-with-keras

In conclusion, in the case of our data, use a multi layer perceptron  should work. Hope kit can help you and have a nice day!